Find a polynomial fit to approximate e^(-2x) = 3x^2 and find an approximation to a root near x=0.

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`e^(-2x)=3x^2`

Let us define f(x) as

`f(x)=e^(-2x)-3x^2`

`f'(x)=-2e^(-2x)-6x`

`f''(x)=4e^(-2x)-6`

`f'''(x)=-8e^(-2x)`

`f^n(x)=(-2)^n e^(-2x)`

Expand in Taylor series about x=0 ,we have

`f(x)=f(0)+xf'(0)+(x^2/2)f''(0)`

`f(x)=0-2x+x^2`

`f(x)=x^2-2x`

`f(0)=0 ,f'(0)=-2,f''(0)=2 , zero`

`Zero`

`of `

`f(x)`

`are `

`x=0`

`and`

`x=2`

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